monte carlo simulation of spin system

8/17/2019 Monte Carlo Simulation of Spin System

http://slidepdf.com/reader/full/monte-carlo-simulation-of-spin-system 1/36

Monte Carlo Simulations of Spin Systems

Wolfhard Janke

Institut fur Physik, Johannes Gutenberg-Universitat, D-55099 Mainz, Germany

Abstract. This lecture gives a brief introduction to Monte Carlo simulations of clas-sical O(n) spin systems such as the Ising (n = 1), XY (n = 2), and Heisenberg (n = 3)model. In the first part I discuss some aspects of Monte Carlo algorithms to generatethe raw data. Here special emphasis is placed on non-local cluster update algorithmswhich proved to be most efficient for this class of models. The second part is de-voted to the data analysis at a continuous phase transition. For the example of thethree-dimensional Heisenberg model it is shown how precise estimates of the transitiontemperature and the critical exponents can be extracted from the raw data. I conclude

with a brief overview of recent results from similar high-precision studies of the Isingand XY model.

1 Introduction

The statistical mechanics of complex physical systems poses many hard problemswhich are difficult to solve by analytical approaches. Numerical simulation tech-niques will therefore be indispensable tools on our way to a better understandingof systems like (spin) glasses, disordered magnets, or proteins, to mention only afew classical problems. Quantum statistical problems in condensed matter or thebroad field of elementary particles and quantum gravity in high-energy physicswould fill many other volumes like this.

The numerical tools can roughly be divided into molecular dynamics (MD)and Monte Carlo (MC) simulations. With the ongoing advances in computertechnology both approaches are expected to gain even more importance thanthey already have today. In the past few years the predictive power of especiallythe MC approach was considerably enhanced by the development of greatlyimproved simulation techniques. Not all of them are already well enough under-stood to be applicable to really complex physical systems. But as a first stepit is gratifying to see that at least for relatively simple spin systems orders of magnitude of computing time can be saved by these refinements. The purposeof this lecture is to give a brief overview on what is feasible today.

From a theoretical view also spin systems are of current interest since onthe one hand they provide the possibility to compare completely different ap-proaches such as field theory, series expansions, and simulations, and on the

other hand they are the ideal testing ground for conceptual considerations suchas universality or finite-size scaling. And last but not least they have found arevival in slightly disguised form in quantum gravity and conformal field theory,where they serve as idealized “matter” fields on Feynman graphs or fluctuatingmanifolds.



2 WolfhardJanke

The rest of the paper is organized as follows. In the next section I first recallthe definition of O(n) spin models and the definition of standard observableslike the specific heat and the susceptibility. Then some properties of phase tran-

sitions are summarized and the critical exponents are defined. In Sec. 3, MonteCarlo methods are described, and Sec. 4 is devoted to an overview of reweightingtechniques. In Sec. 5, applications to the three-dimensional classical Heisenbergmodel are discussed, and in Sec. 6, I conclude with a few comments on similarsimulations of the Ising and XY model.

2 Spin Models and Phase Transitions

2.1 Models and Observables

In the following we shall confine ourselves to O(n) symmetric spin models whosepartition function is defined as

Z n(β ) =σi

exp(−βH n) , (1)

with

H n = −J ij

σi · σj ; σi = (σ(1)i , σ

(2)i , . . . , σ

(n)i ); |σi| = 1 . (2)

Here β = 1/kBT is the inverse temperature, the spins σi live on the sites i of a D-dimensional cubic lattice of volume V = LD, and the symbol ij indicatesthat the lattice sum runs over all 2D nearest-neighbor pairs. We always assumeperiodic boundary conditions.

Standard observables are the internal energy per site, e = E/V , with E =

−d ln Z n/dβ ≡ H n, and the specific heat,

C/kB = de

d(kBT ) = β 2

H 2n − H n2

/V . (3)

On finite lattices the magnetization and susceptibility are usually defined as

m = M/V = |σav|; σav =i

σi/V , (4)

χ = βV σ2

av− |σav|2

. (5)

In the high-temperature phase one often employs the fact that the magnetizationvanishes in the infinite volume limit and defines

χ′ = βV σ2av . (6)

Similarly, the spin-spin correlation function can then be taken as

G(xi − xj) = σi · σj . (7)



Monte Carlo Simulations of Spin Systems 3

At large distances, G(x) decays exponentially and the correlation length ξ canbe defined as

ξ =

− lim|x|→∞ |

x

|/ ln G(x) . (8)

For n = 1, the partition function Z n(β ) describes the standard Ising model

where the spins can take only the two discrete values, σi ≡ σ(1)i = ±1. For

n = 2, 3, . . . the spins vary continuously on the n-dimensional unit sphere. Par-ticularly thoroughly studied cases are n = 2 (XY model) and n = 3 (Heisenbergmodel). The limit n −→ ∞ is known to be equivalent to the spherical model of Berlin and Kac (1952). In three dimensions (3D) the model exhibits for all na continuous phase transition from an ordered low-temperature phase to a dis-ordered high-temperature phase. The associated critical exponents are genericfor the so-called O(n) universality classes. In two dimensions (2D) the situationis little more complex. For the 2D Ising model the exact solution by Onsagerand later Yang predicts a continuous order-disorder phase transition similar to

3D. For all n ≥ 2, however, the spin degrees of freedom are continuous and,as a consequence of the Mermin-Wagner-Hohenberg theorem, the magnetizationvanishes for all temperatures. The 2D XY model nevertheless displays a verypeculiar (infinite order) Kosterlitz-Thouless transition (Kosterlitz and Thouless1973, Berezinskii 1971, Kosterlitz 1974). Due to the O(2) symmetry this modeladmits point like topological defects (vortices) which are tightly bound to pairsat low temperatures. With increasing temperature isolated vortices are entrop-ically favored, and the transition is usually pictured as the point where vortexpairs start to dissociate (for a review see, e.g., Kleinert (1989a)). For the 2DHeisenberg model and all other 2D O(n) models with n > 3, on the other hand,it is commonly believed that there is no phase transition at finite temperature.1

For later reference we also recall another generalization of the Ising model,the q -state Potts model (Potts 1952) with Hamiltonian

H Potts = −J ij

δ σiσj ; σi ∈ 1, . . . , q . (9)

This generalization has in 3D for all q ≥ 3 a first-order transition; and in 2D itis exactly known to exhibit a second-order transition for q ≤ 4, and a first-ordertransition for all q ≥ 5 (Wu 1982, 1983).

2.2 Phase Transitions

In limiting cases like low and high temperatures (or fields, pressure, etc.) thephysical degrees of freedom usually decouple and the statistical mechanics of

even complex systems becomes quite manageable. Much more interesting is theregion in between these extremes where strong cooperation effects may causephase transitions, e.g., from an ordered phase at low temperatures to a disordered

1 For an alternative view see, however, Patrascioiu and Seiler (1995) and references toearlier work therein.



4 WolfhardJanke

phase at high temperatures. To predict the properties of this most difficult regionof a phase diagram as accurately as possible is the most challenging objective of all statistical mechanics approaches, including numerical simulations.

The theory of phase transitions is a very broad subject described comprehen-sively in many textbooks. Here we shall be content with a rough classification in

first-order and second-order (or, more generally, continuous) phase transitions,and a summary of those properties that are most relevant for numerical simula-tions. Some characteristic properties of first- and second-order phase transitionsare sketched in Fig. 1.

0 1

T/T0

0

1

m

First-Order Transition

∆m

0 1

T/T0

C,χ

Cd

Co

0 1

T/Tc

0

1

m

Second-Order Transition

∝(1-T/Tc)β

0 1

T/Tc

C,χ

∝|1-T/Tc|-α∝(1-T/Tc)-α

Fig. 1. The characteristic behaviour of the magnetization, m, specific heat, C , andsusceptibility, χ, at first- and second-order phase transitions.

Most phase transitions in Nature are of first order (Gunton et al. 1983, Binder1987, Herrmann et al. 1992, Janke 1994). The best known example is the field-driven transition in magnets at temperatures below the Curie point, while theparadigm of a temperature-driven first-order transition experienced every day




is ordinary melting (Kleinert 1989b, Janke and Kleinert 1986). In general, first-order phase transitions are characterized by discontinuities of the order param-eter (the jump ∆m of the magnetization m in Fig. 1), or the energy (the latent

heat ∆e), or both. This reflects the fact that, at the transition temperature T 0,two (or more) phases can coexist. In the example of a magnet at low temperaturesthe coexisting phases are the phases with positive and negative magnetization,while at the melting transition they are the solid (ordered) and liquid (disor-dered) phases. The correlation length in the coexisting pure phases is finite.2

Consequently also the specific heat, C , and the susceptibility, χ, do not divergein the pure phases. Mathematically there are, however, superimposed delta func-tion like singularities associated with the jumps of e and m.

In this lecture we will mainly consider second-order phase transitions, whichare characterized by a divergent correlation length ξ at the transition tempera-ture T c ≡ 1/β c. The growth of correlations as one approaches the critical regionfrom high temperatures is illustrated in Fig. 2, where six typical configurations

of the 2D Ising model at inverse temperatures β/β c = 0.50, 0.70, 0.85, 0.90,0.95, and 0.98 are shown. Because for an infinite correlation length fluctuationson all length scales are equally important, one expects power-law singularitiesin thermodynamic functions. The leading singularity of the correlation length isusually parametrized in the high-temperature phase as

ξ = ξ 0+ |1 − T /T c|−ν + . . . (T ≥ T c) , (10)

where the . . . indicate subleading corrections (analytical as well as confluent).This defines the critical exponent ν and the critical amplitude ξ 0+ on the high-temperature side of the transition. In the low-temperature phase one expects asimilar behaviour,

ξ = ξ 0−(1 − T /T c)−ν + . . . (T ≤ T c) , (11)

with the same critical exponent ν but a different critical amplitude ξ 0− = ξ 0+ .An important feature of second-order phase transitions is that due to the

divergence of ξ the short-distance details of the Hamiltonian should not matter.This is the basis of the universality hypothesis which states that all systems withthe same symmetries and same dimensionality should exhibit similar singularitiesgoverned by one and the same set of critical exponents. For the amplitudes thisis not true, but certain amplitude ratios are also universal.

The singularities of the specific heat, magnetization, and susceptibility aresimilarly parametrized by the critical exponents α, β , and γ , respectively,

C = C reg + C 0|1 − T /T c|−α + . . . , (12)

m = m0(1

−T /T c)β + . . . , (13)

χ = χ0|1 − T /T c|−γ + . . . , (14)

2 For the 2D q -state Potts model with q ≥ 5, where many exact results are known, thisis illustrated by the recent simulations of Janke and Kappler (1994, 1995a, 1995b,1995c, 1995d); for details see Kappler (1995).



6 WolfhardJanke

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

Fig. 2. From high temperatures (upper left) to the critical region (lower right), char-acterized by large spatial correlations. Shown are actual 2D Ising configurations for a100 × 100 lattice at β /β c = 0.50, 0.70, 0.85, 0.90, 0.95, and 0.98.




where C reg is a regular background term, and the amplitudes are again differenton the two sides of the transition, cp. Fig. 1.

Finite-Size Scaling. For systems of finite size, as in any numerical simula-tion, the correlation length cannot diverge, and also the divergencies in all otherquantities are then rounded and shifted. This is illustrated in Fig. 3, where thespecific heat of the 2D Ising model on various L×L lattices is shown. The curvesare computed from the exact solution of Ferdinand and Fisher (1969) for anyLx × Ly lattice with periodic boundary conditions.

0.2 0.3 0.4 0.5 0.6 0.7

β

0

1

2

3

s p e c i f i c h e a t

L=128

64

32

16

8

4

2D Ising

periodic b.c.

βc

Fig. 3. Finite-size scaling behaviour of the specific heat of the 2D Ising model on L × Llattices. The critical point is indicated by the arrow on the top axis.

Near T c the role of ξ in the scaling formulas is then taken over by the linearsize of the system, L. By writing |1−T /T c| ∝ ξ −1/ν −→ L−1/ν , we see that at T cthe scaling laws (12)-(14) are replaced by the finite-size scaling (FSS) Ansatze,

C = C reg + aLα/ν + . . . , (15)

m ∝ L−β/ν + . . . , (16)

χ ∝ Lγ/ν + . . . . (17)

More generally these scaling laws are valid in the vicinity of T c as long as thescaling variable x = (1 − T /T c)L1/ν is kept fixed (Binder 1979, Barber 1983,Privman 1990, Binder 1992b). In particular this is true for the locations T max of the (finite) maxima of thermodynamic quantities, which are expected to scalewith the system size as T max = T c(1 − xmaxL−1/ν + . . .). In this more general



8 WolfhardJanke

formulation the scaling law for, e.g., the susceptibility reads χ(T, L) = Lγ/ν f (x).By plotting χ(T, L)/Lγ/ν vs the scaling variable x, one thus expects that thedata for different T and L fall onto a kind of master curve. While this is a nice

way to demonstrate the scaling properties qualitatively, it is not particularlysuited for quantitative analyses.

Since the goal of most simulation studies of spin systems are high-precisionestimates of the critical temperature and the critical exponents, one thereforeprefers fits either to the “thermodynamic” scaling laws (12)-(14) or to the FSSAnsatze (15)-(17).

Similar considerations for first-order phase transitions show that also thedelta function like singularities, originating from the phase coexistence, aresmeared out for finite systems (Fisher and Berker 1982, Privman and Fisher1983, Binder and Landau 1984, Challa et al. 1986, Privman and Rudnik 1990).In finite systems they are replaced by narrow peaks whose height (width) growsproportional to the volume (1/volume) (Borgs and Kotecky 1990, 1992, Lee and

Kosterlitz 1990, 1991, Borgs et al. 1991, Borgs and Janke 1992, Janke 1993).

3 The Monte Carlo Method

Let us now discuss how the expectation values in (3)-(7) can be computed numer-ically. A direct summation of the partition function is impossible, since alreadyfor the Ising model with only two possible states per site the number of termswould be enormous: 22500 ≈ 10753 for a 50 × 50 lattice! Also a naive randomsampling of the spin configurations does not work. Here the problem is thatthe relevant region in the high-dimensional phase phase is relatively narrow andhence too rarely hit by random sampling. The solution to this problem is knownsince long. One has to use the importance sampling technique (Hammersley and

Handscomb 1965) which is designed to draw configurations according to theirBoltzmann weight,P eq[σi] ∝ exp(−βH [σi]) . (18)

In more mathematical terms one sets up a Markov chain,

. . . W −→ σi W −→ σ′i W −→ σ′′i W −→ . . . ,

with a transition operator W satisfying the conditions

a) W (σi−→σ′i) ≥ 0 for all σi, σ′i , (19)

b)σ′

i

W (σi−→σ′i) = 1 for all σi , (20)

c) σi

W (

σi

−→σ′i

)P eq[

σi

] = P eq[

σ′i

] for all

σ′i

. (21)

From (21) we see that P eq is a fixed point of W . A somewhat simpler sufficientcondition is detailed balance,

P eq[σi]W (σi−→σ′i) = P eq[σ′i]W (σ′i−→σi) . (22)




After an initial equilibration time, expectation values can then be estimated asan arithmetic mean over the Markov chain, e.g.,

H =σi

H [σi]P eq[σi] ≈ 1N

N j=1

H [σi]j . (23)

A more detailed exposition of the basic concepts underlying any Monte Carloalgorithm can be found in many textbooks and reviews (Binder 1979, Binderand Heermann 1988, Heermann 1990, Binder 1992a).

3.1 Estimators and Autocorrelation Times

In principle there is no limitation on the choice of observables. The expectationvalue O of any observable O can be estimated in a MC simulation as a simple

arithmetic mean over the Markov chain, ¯

O= 1

N N

j=1 Oj , where

Oj

≡ O[

σi

]jis the measurement after the j-th iteration. Conceptually it is important todistinguish between the expectation value O, which is an ordinary number, andthe estimator O, which is a random number fluctuating around the theoreticallyexpected value. Of course, in practice one does not probe the fluctuations of the estimator directly (which would require repeating the whole MC simulationmany times), but rather estimates its variance σ2

O = [ O − O]2 = O2−O2

from the distribution of Oj . If the N measurements Oj were all uncorrelatedthen the relation would simply be σ2

O = σ2

Oj/N , with σ2

Oj= O2

j − Oj2. Forcorrelated measurements one obtains after some algebra

σ2O =

σ2Ok

N 2τ O,int , (24)

where the integrated autocorrelation time,

τ O,int = 1

2 +

N j=1

A( j)

1 − j

N

, (25)

turns out to be a sum (“integral”) over the the autocorrelation function,

A( j) = OiOi+j− OiOi

O2i − OiOi . (26)

For large time separations the autocorrelation function decays exponentially,

A( j) j→∞

−→ ae−j/τ O,exp , (27)

with a being a constant. This defines the exponential autocorrelation time τ O,exp.Due to the exponential decay of A( j), in any meaningful simulation with N ≫τ O,exp, the correction term in parentheses in (25) can safely be neglected. Noticethat only if A( j) is a pure exponential, the two autocorrelation times, τ O,int and



10 WolfhardJanke

τ O,exp, coincide (up to minor corrections for small τ O,int) (Madras and Sokal1988).

The important point is that for correlated measurements the statistical er-

ror ǫO ≡ σ2O

on the MC estimator O is enhanced by a factor of

2τ O,int.

This can be rephrased by writing the statistical error similar to the uncorre-

lated case as ǫO =

σ2Oj

/N eff , but now with an effective statistics parameter

N eff = N/2τ O,int. This shows more clearly that only every 2τ O,int iteration themeasurements are approximately uncorrelated and gives a better idea of therelevant effective size of the statistical sample. Since some quantities (e.g., thespecific heat or susceptibility) can severely be underestimated if the effectivestatistics is too small (Ferrenberg et al. 1991), any serious simulation shouldtherefore provide at least a rough order-of-magnitude estimate of autocorrela-tion times.

Unfortunately, it is very difficult to give reliable a priori estimates, and an

accurate numerical analysis is often too time consuming (as a rough estimate itis about ten times harder to get precise information on dynamic quantities thanon static quantities like critical exponents). To get at least an idea of the ordersof magnitude, it is useful to record the “running” autocorrelation time

τ O,int(k) = 1

2 +

kj=1

A( j) , (28)

which approaches τ O,int for large k. Approximating the tail end of A( j) by asingle exponential as in (27), one derives (Janke and Sauer 1995)

τ O,int(k) = τ O,int − a

e1/τ O,exp

−1

e−k/τ O,exp . (29)

The latter expression may be used for a numerical estimate of both the expo-nential and integrated autocorrelation times.

To summarize this subsection, any realization of a Markov chain (i.e., MCupdate algorithm) is characterized by autocorrelation times which enter directlyin the statistical errors of MC estimates. Since correlations always increase thestatistical errors, it is a very important issue to develop MC update algorithmsthat keep autocorrelation times as small as possible. In the next subsectionwe first discuss the classical Metropolis algorithm as an example of an updatealgorithm which near criticality is plagued by huge temporal correlations. Thediscussion of cluster updates in the next subsection then demonstrates that thereindeed exist clever ways to overcome this critical slowing down problem.

3.2 Metropolis Algorithm

In the standard Metropolis algorithm (Metropolis et al. 1953) the Markov chainis realized by local updates of single spins. If E and E ′ denote the energy before




and after the spin flip, respectively, then the probability to accept the proposedspin update is given by

W (σi−→σ′i) =

exp[−β (E ′ − E )] E ′ ≥ E 1 E ′ < E .

(30)

If the energy is lowered, the spin flip is always accepted. But even if the energyis increased, the flip has to be accepted with a certain probability to ensurethe proper treatment of entropic contributions. In thermal equilibrium the free

energy is minimized and not the energy. Only at zero temperature (β −→ ∞) thisprobability tends to zero and the MC algorithm degenerates to a minimizationalgorithm for the energy functional. With some additional refinements, this isthe basis of the simulated annealing technique also discussed in this volume,which is usually applied to hard optimization and minimization problems.

There are many ways how to choose the spins to be updated. The lattice sites

may be picked at random or according to a random permutation, which can beupdated every now and then. But also a simple fixed lexicographical order ispermissible. Or one updates first all odd and then all even sites, which is theusual choice in vectorized codes. A so-called lattice sweep is completed when onthe average3 for all spins an update was proposed.

The advantage of this simple algorithm is its flexibility which allows the ap-plication to a great variety of physical systems. The great disadvantage, however,is that this algorithm is plagued by large autocorrelation times, as most otherlocal update algorithms (one exception is the overrelaxation method (Creutz1987, Adler 1988, Neuberger 1988, Gupta et al. 1988)). Empirically one finds thatthe autocorrelation time grows proportional to the spatial correlation length,

τ ∝ ξ z

, (31)

with a dynamical critical exponent z ≈ 2. Heuristically this can be understoodby assuming that local excitations diffuse through the system like a random walk.Since ξ diverges at criticality, the Metropolis algorithm thus severely suffers fromcritical slowing down . Of course, in finite systems ξ cannot diverge. Then ξ isreplaced by the linear lattice size L, yielding τ ∝ Lz.

The problem of critical slowing down can be overcome by non-local updatealgorithms. In the past few years several different types of such algorithms havebeen proposed. Quite promising results were reported with Fourier acceleration(Doll et al. 1985) and multigrid techniques (Goodman and Sokal 1986, 1989,Mack 1988, Kandel et al. 1988, 1989, Mack and Meyer 1990). A very nice andpedagogical introduction to these techniques is given by Sokal (1989, 1992).For the O(n) spin models considered here, however, the best performance wasachieved with cluster update algorithms which will be described in the nextsubsection in more detail.

3 This is only relevant in the case where the lattice sites are picked at random.



12 WolfhardJanke

3.3 Cluster Algorithms

As we shall see below, cluster update algorithms (Swendsen and Wang 1987,

Wolff 1989a) are much more powerful than the Metropolis algorithm. Unfortu-nately, however, they are less general applicable. We therefore consider first onlythe Ising model, where the prescription for cluster update algorithms can easilybe read off from the equivalent Fortuin-Kasteleyn representation (Kasteleyn andFortuin 1969, Fortuin and Kasteleyn 1972, Fortuin 1972a, 1972b),

Z =σi

exp

β

ij

σiσj

(32)

=σi

ij

eβ

(1 − p) + pδ σiσj

(33)

= σi

nij

ij

eβ (1

− p)δ nij,0 + pδ σiσjδ nij,1 , (34)

with p = 1 − e−2β . (35)

Here the nij are bond variables which can take the values nij = 0 or 1, interpretedas “deleted” or “active” bonds. In the first line we used the trivial fact thatthe product σiσj of two Ising spins can only take the two values ±1, so thatexp(βσiσj) = x + yδ σiσj can easily be solved for x and y. And in the second line

we made use of the “deep” identity a + b =1

n=0 (aδ n,0 + bδ n,1).

Swendsen-Wang Cluster. According to (34) a cluster update sweep thenconsists of alternating updates of the bond variables nij for given spins withupdates of the spins σi for a given bond configuration. In practice one proceedsas follows:

1. Set nij = 0 if σi = σj, or assign values nij = 1 and 0 with probability p and1 − p, respectively, if σi = σj , cp. Fig. 4.

2. Identify clusters of spins that are connected by “active” bonds (nij = 1).3. Draw a random value ±1 independently for each cluster (including one-site

clusters), which is then assigned to all spins in a cluster.

Technically the cluster identification part is the most complicated step, but thereare by now quite a few efficient algorithms available which can even be used onparallel computers. Vectorization, on the other hand, is only partially possible.

Notice the difference between the just defined stochastic clusters and geo-

metric clusters whose boundaries are defined by drawing lines through bondsbetween unlike spins. In fact, since in the stochastic cluster definition also bondsbetween like spins are “deleted” with probability p0 = 1− p = exp(−2β ), stochas-tic clusters are on the average smaller than geometric clusters. Only at zero tem-perature (β −→ ∞) p0 approaches zero and the two cluster definitions coincide.




nij=0 n

ij=1 n

ij=0

always p1=p p

0=1-p

Fig. 4. Illustration of the bond variable update. The bond between unlike spins isalways “deleted” as indicated by the dashed line. A bond between like spins is only“active” with probability p = 1 − exp(−2β ). Only at zero temperature (β −→ ∞)stochastic and geometric clusters coincide.

As described above, the cluster algorithm is referred to as Swendsen-Wang (SW)or multiple-cluster update (Swendsen and Wang 1987). The distinguishing pointis that the whole lattice is decomposed into stochastic clusters whose spins areassigned a random value +1 or −1. In one sweep one thus attempts to updateall spins of the lattice.

Wolff Cluster. Shortly after the original discovery of cluster algorithms, Wolff (1989a) proposed a somewhat simpler variant in which only a single cluster isflipped at a time. This variant is therefore sometimes also called single-clusteralgorithm. Here one chooses a lattice site at random, constructs only the clusterconnected with this site, and then flips all spins of this cluster. A typical ex-

ample is shown in Fig. 5. In principle, one could also here choose a value +1 or−1 at random, but then nothing at all would be changed if one hits the currentvalue of the spins. Here a sweep consists of V /C single cluster steps, whereC denotes the average cluster size. With this definition autocorrelation timesare directly comparable with results from the Metropolis or Swendsen-Wang al-gorithm. Apart from being somewhat easier to program, Wolff’s single-clustervariant is usually more efficient than the Swendsen-Wang multiple-cluster algo-rithm, especially in 3D. The reason is that with the single-cluster method, onthe average, larger clusters are flipped.

Embedded Cluster. While it is quite easy to generalize the derivation (32) –(35) to q -state Potts models, for the O(n) spin models with n

≥ 2 one needs a

new strategy (Wolff 1989a, 1989b, 1990, Hasenbusch 1990). Here the basic ideais to isolate Ising degrees of freedom by projecting σi onto a randomly chosenunit vector r,

σi = σi + σ

⊥i ; σ

i = ǫ |σi · r| r; ǫ = sign(σi · r) . (36)



14 WolfhardJanke

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

0.0 20.0 40.0 60.0 80.0 100.00.0

20.0

40.0

60.0

80.0

100.0

Fig. 5. Illustration of the Wolff cluster update. Upper left: Initial configuration. Upperright: The stochastic cluster is marked. Lower left: Final configuration after flippingthe spins in the cluster. Lower right: The flipped cluster. The shown spin configurationis from an actual simulation of the 2D Ising model at 0.97 × β c on a 100 × 100 lattice.

If this is inserted in the original Hamiltonian one ends up with an effectiveHamiltonian

H = −ij

J ijǫiǫj + const , (37)

with positive random couplings,

J ij = J |σi · r||σj · r| ≥ 0 , (38)

whose Ising degrees of freedom ǫi can be updated with a cluster algorithm asdescribed above.




For O(n) spin models the performance of both types of cluster algorithmsis excellent. As is demonstrated in Table 1 and Fig. 6, critical slowing down isdrastically reduced. We see that especially in three dimensions the Wolff cluster

algorithm performs better than the i Swendsen-Wang algorithm. Compared withthe Metropolis algorithm, factors of up to 10 000 in CPU time have been savedin realistic simulations (Baillie 1990, Swendsen et al. 1992).

Table 1. Dynamical critical exponents z for the 2D and 3D Ising model (τ ∝ Lz)

algorithm D=2 D=3 observable authors

Metropolis 2.125 2.03

Swendsen-Wang cluster 0.35(1) 0.75(1) z E,exp Swendsen and Wang (1987)

0.27(2) 0.50(3) z E,int Wolff (1989c)

0.20(2) 0.50(3) z χ,int Wolff (1989c)0(log L) – z M,exp Heermann and Burkitt (1990)

0.25(5) – z M,rel Tamayo (1993)

Wolff cluster 0.26(2) 0.28(2) z E,int Wolff (1989c)

0.13(2) 0.14(2) z χ,int Wolff (1989c)

0.25(5) 0.3(1) z E,rel Ito and Kohring (1993)

Improved Estimators. A further advantage of cluster algorithms is that they

lead quite naturally to so-called improved estimators which are designed to fur-ther reduce the statistical errors. Suppose we want to measure the expectationvalue O of an observable O. Then any estimator O satisfying O = O ispermissible. This does not determine O uniquely since there are infinitely manyother possible choices, O′ = O + X , where the added estimator X has zero ex-pectation, X = 0. The variances of the estimators O′, however, can be quitedifferent and are not necessarily related to any physical quantity (contrary tothe standard mean-value estimator of the energy whose variance is proportionalto the specific heat). It is exactly this freedom in the choice of O which allowsthe construction of improved estimators.

For the single-cluster algorithm an improved “cluster estimator” for the spin-spin correlation function in the high-temperature phase, G(xi − xj) ≡ σi ·σj,is given by (Wolff 1990)

G(xi − xj) = n V

|C |r · σi r · σjΘC (xi)ΘC (xj) , (39)

where r is the normal of the mirror plane used in the construction of the clusterof size |C | and ΘC (x) is its characteristic function (=1 if x ∈ C and 0 other-



16 WolfhardJanke

10 100

L

1

10

τint

3D XY model

τχ=4.2 L0.31

τe=3.1 L0.46

SW

τe=1.3 L

0.25

τχ=1.7

Wolff

Fig. 6. Double logarithmic plot of the integrated autocorrelation times for the Swend-sen-Wang (SW) and Wolff algorithm of the 3D XY model near criticality. The squares(β = 0.45421) are taken from Hasenbusch and Meyer (1990), the circles (β = 0.4539)and diamonds (β = 0.4543) from Janke (1990).

wise). For the Fourier transform, G(k) =

xG(x) exp(−ik · x), this implies the

improved estimator

ˆG(k) = n

|C

|

i∈C

r · σi cos kxi

2

+

i∈C

r · σi sin kxi

2 , (40)

which, for k = 0, reduces to an improved estimator for the susceptibility χ′ inthe high-temperature phase,

ˆG(0) = χ′/β = n

|C |

i∈C

r · σi2

. (41)

For the Ising model (n = 1) this reduces to χ′/β = |C |, i.e., the improvedestimator of the susceptibility is just the average cluster size of the single-clusterupdate algorithm. For the XY and Heisenberg model one finds empirically that intwo as well as in three dimensions

|C

| ≈0.81χ′/β for n = 2 (Wolff 1989b, Janke

1990) and |C | ≈ 0.75χ′/β for n = 3 (Wolff 1990, Holm and Janke 1993b),respectively.

It should be noted that by means of the estimators (39)-(41) a significantreduction of variance should only be expected outside the FSS region where theaverage cluster size is small compared with the volume of the system.




3.4 Multicanonical Algorithms for First-Order Transitions

Let us finally make a few brief comments on numerical simulations of first-orderphase transitions (Janke 1994). Since here the correlation lengths in the purephases are finite, the numerical problems are completely different than in thecase of second-order phase transitions. Here the origin of numerical difficultiesnear the transition point can be traced back to the coexistence of two phaseswhich, for finite systems, is reflected by a double-peak structure of the corre-sponding order-parameter or energy distribution. The minimum between the twopeaks is governed by mixed phase configurations which are strongly suppressedby an additional Boltzmann factor ∝ exp(−2σLD−1). Here σ denotes the inter-face tension at the phase boundaries, LD−1 is the cross-section of the (cubic)system of size V = LD, and the factor 2 takes into account the usually employed

periodic boundary conditions (Binder 1981, 1982). The problem of numericalsimulations is to achieve equilibrium between the two phases. Most of the timethe system spends in the pure phases. Only very rarely it “tunnels” throughthe exponentially suppressed mixed-phase region from one phase to the other.These rare tunneling events, however, are necessary to achieve equilibrium be-tween the pure phases. The relevant time scale of equilibrium simulations is thusgiven by the inverse of the additional Boltzmann factor, i.e., the characteristictime τ grows exponentially with the system size, τ ∝ exp(2σLD−1) (Billoire etal. 1991). Since for an accurate numerical study the simulation (and thus com-puting) time must be much larger than τ , this phenomenon has been termed theexponential or super-critical slowing down problem.

A surprisingly simple solution to this problem was discovered by Berg and

Neuhaus (1991). In what they call “multicanonical” simulations one determinesiteratively artificial weight factors which modify the original Hamiltonian insuch a way that the order-parameter or energy distribution is approximatelyflat between the two peaks of the canonical distribution. Since then the systemhas no longer to pass through an exponentially suppressed region one expectsin multicanonical simulations a drastic reduction of the characteristic time scaleτ . A simple random walk argument suggests a power-law behaviour, τ ∝ V α,which has indeed been confirmed in numerical simulations of 2D Potts models(Berg and Neuhaus 1992, Janke et al. 1992, Rummukainen 1993).

The multicanonical technique is strictly speaking not an update algorithmbut a reweighting procedure as discussed in detail in the next section. In prin-ciple, it can therefore be combined with any legitimate update algorithm. The

earlier studies all employed the Metropolis or heat-bath algorithm. In more re-cent work it was shown that also combinations with multigrid techniques (Jankeand Sauer 1994, 1995, Sauer 1994) and cluster update algorithms (Rummukainen1993, Janke and Kappler 1995e, 1995f, Kappler 1995) are feasible and can furtherreduce autocorrelation times.



18 WolfhardJanke

4 Reweighting Techniques

Even though the physics underlying reweighting techniques is extremely sim-

ple and the basic idea has been known since long (see the list of references byFerrenberg and Swendsen (1989a)), their power in practice has been realizedonly quite recently (Ferrenberg and Swendsen 1988, 1989a). The best perfor-mance is achieved near criticality, and in this sense reweighting techniques arecomplementary to improved estimators.

If we denote the number of states (spin configurations) that have the sameenergy by Ω (E ), the partition function at the simulation point β 0 can always bewritten as4

Z (β 0) =E

Ω (E )e−β0E . (42)

This shows that the energy distribution P β0(E ) (normalized to unit area) isgiven by

P β0(E ) = Ω (E )e−β0E /Z (β 0) . (43)

It is then easy to see that, given P β0(E ), the energy distribution is actuallyknown for any β ,

P β(E ) = c P β0(E )e−(β−β0)E , (44)

where c is a normalization constant (which in practice is trivially determinedby enforcing the condition

E P β(E ) = 1. Formally, one easily finds that c =

1/

E P β0(E )e−(β−β0)E = Z (β 0)/Z (β ).) Knowing P β(E ), expectation values of the form f (E ) are easy to compute,

f (E )(β ) =E

f (E )P β(E ) . (45)

Since the relative statistical errors increase in the wings of P β0(E ) one expects(44), (45) to give reliable results only for β near β 0. If β 0 is near criticality, thedistribution is relatively broad and the method works best. In this case reliableestimates from (44) can be expected for β values in an interval around β 0 of width∝ L−1/ν , i.e., just in the FSS region. As a rule of thumb the reweighting rangecan be determined by the condition that the peak location of the reweighteddistribution should not exceed the energy values at which the input distributionhad decreased to one third of its maximum value (Alves et al. 1992). In mostapplications this range is wide enough to locate from a single simulation, e.g.,the specific-heat maximum by using any standard maximization routine.

This is illustrated in Figs. 7 and 8, again for simplicity for the 2D Isingmodel. In Fig. 7 the filled circle shows the result of a MC simulation at β c =

log(1 + √ 2)/2 ≈ 0.440686 . . ., using the Swendsen-Wang cluster algorithm with5000 sweeps for equilibration and 50 000 sweeps for measurements. The results

4 For simplicity we consider here only models with discrete energies. If the energyvaries continuously, sums have to be replaced by integrals, etc.. Also lattice sizedependencies are suppressed to keep the notation short.




0.3 0.4 0.5 0.6

β

0

1

2

s p e c i f i c h e a t

2D Ising

162

Fig. 7. Specific heat of the 2D Ising model computed by reweighting () from a singleMC simulation at β 0 = β c (•). The continuous line shows for comparison the exactresult of Ferdinand and Fisher (1969).

0.5 1.0 1.5 2.0

-e

0

10

c o u n t s

β=0.375β=0.475

β0=β

c

Fig. 8. The energy histogram at the simulation point β 0 = β c, and reweighted toβ = 0.375 and β = 0.475. The black dots show histograms obtained in additionalsimulations at these temperatures.

of the reweighting procedure are shown as open circles (recall that the spacingbetween the circles can be made as small as desired, here it was chosen quite large

for clarity of the plot) and compared with the exact curve (Ferdinand and Fisher1969). We see that even with this rather modest statistics the whole specific-heatpeak can be obtained with reasonable accuracy from a single simulation. But wealso notice significant deviations in the tails of the peak. To understand the originof the deviations it is useful to have a look at the energy histograms in Fig. 8. The



20 WolfhardJanke

curve labeled β 0 = β c is the histogram of the MC data at the simulation point,and the other two histograms at β = 0.375 and β = 0.475 are computed fromthis input histogram by reweighting. For comparison we have also included the

histograms obtained from additional MC simulations (with the same statistics)at the two β -values, indicated by the black dots. We see that the reweightedhistogram at β = 0.475 looks smooth to the eye - and indeed agrees very wellwith the “direct” result of the additional MC simulation at this temperature.In Fig. 7 this is reflected by the still very good agreement of the numerical andthe exact result. For β = 0.375, on the other hand, already visually one wouldnot trust the reweighted histogram. While the tail on the right-hand side is stillin reasonable agreement with the “direct” simulation, the left tail is obviouslyhopelessly wrong. By recalling that the reweighted histograms are computed bymultiplying the input histogram with exponential factors, this is no surprise atall: For −e <∼ 1 there are hardly any entries in the input histogram and hencethe relative statistical errors (∝ 1/

√ counts) are huge. This is the source for the

large deviations from the exact curve in Fig. 7 for β <∼ 0.4.The information stored in P β0(E ) is not yet sufficient to calculate also the

magnetic moments mk(β ) as a function of β from a single simulation at β 0.Conceptually, the simplest way to proceed is to record the two-dimensional his-togram P β0(E, M ), where M = mV is the total magnetization. Because of diskspace limitations one sometimes prefers to measure instead the “microcanonicalaverages”

mk(E ) =M

P β0(E, M )mk/P β0(E ) , (46)

where the relation

M P β0(E, M ) = P β0(E ) was used. In practice one simplyaccumulates the measurements of mk in different slots or bins according to the

energy of the configuration and normalizes at the end by the total number of hits of each energy bin. Clearly, once mk(E ) is determined, this is a specialcase of f (E ) in (45), so that

mk(β ) =

E mk(E )P β0(E )e−(β−β0)E

E P β0(E )e−(β−β0)E . (47)

Similar to Ω (E ) in (42), theoretically also the microcanonical averages mk(E )do not depend on the temperature at which the simulation is performed. Due tothe limited statistics in the wings of P β0(E ), however, there is only a finite rangearound E 0 ≡ E (β 0) where one can expect reasonable results for mk(E ).Outside of this range it simply can happen (and does happen) that there are noevents to be averaged. This is illustrated in Fig. 9, where

m2

(E ) is plotted for

the 3D Heisenberg model as obtained from three runs at different temperatures.We see that the function looks smooth only in the range where the statisticsof the corresponding energy histogram is high enough. To take full advantageof the histogram reweighting technique it is therefore advisable to perform afew simulations at slightly different inverse temperatures β i. Instead of spending




1.96 1.98 2.00 2.02 2.04

E/V

0

10

20

30

40

50

60

70

P β

( E )

β=0.6929

β=0.6914β=0.6941

L = 48

1.96 1.98 2.00 2.02 2.04

E/V

0.00

0.01

0.02

0.03

0.04

0.05

< < m

2 > > ( E )

β = 0.6941

β = 0.6929

β = 0.6914

Fig. 9. Energy histograms and microcanonical magnetization squared of the 3D Heisen-berg model at the three simulation points for L = 48.

all computer time in a single long run, it is usually more efficient to performthree or four shorter runs. To find the best solution, however, is a very difficultoptimization problem, which depends on many details of the model under study!

Now the question arises how to combine the data from different runs mostefficiently. A very clear way is to compute for each simulation (at β i) the β -dependence of Oi ≡ O(βi)(β ) plus the associated statistical error ∆Oi, using jack-knife techniques (Miller 1974, Efron 1982), say. A single optimal expression



22 WolfhardJanke

for O ≡ O(β ) is then obtained by combining the values Oi in such a way thatthe relative error ∆O/O is minimized (Holm and Janke 1993b),

O =

O1

(∆O1)2 +

O2

(∆O2)2 +

O3

(∆O3)2

(∆O)2 , (48)

with

1

(∆O)2 =

1

(∆O1)2 +

1

(∆O2)2 +

1

(∆O3)2 . (49)

A different procedure at the level of distribution functions was discussed byFerrenberg and Swendsen (1989b). For the specific heat the two methods werefound (Holm and Janke 1993b) to give comparable results within the statisticalerrors. The optimization at the level of observables, however, is simpler to apply

to quantities involving constant energy averages such as m(E ), and, moreimportantly, minimizes the error on each observable of interest separately.

5 Applications to the 3D Heisenberg Model

Let us now turn to applications of the just described techniques to the 3D clas-sical Heisenberg model (n = 3), focussing on an accurate determination of thetransition temperature and the critical exponents. To this end the cluster up-date algorithm proved to be a very important tool. Previous studies (Peczakand Landau 1990a, 1990b, 1993, Peczak et al. 1991) employing the Metropolisalgorithm reported for the magnetization an exponential autocorrelation timeof τ m,exp = aLz, with a ≈ 3.76 and z = 1.94(6). In simulations with the single-cluster algorithm we obtained for the susceptibility values of τ χ,int ≈ 1.5 − 2.0(Holm and Janke 1993a, 1993b, 1994). As for the 3D Ising and 3D XY modelcritical slowing down is thus almost completely eliminated. Compared to theMetropolis algorithms this implies for a 803 lattice an acceleration of the simu-lation by about four orders of magnitude.

In the paper by Holm and Janke (1993b) two sets of MC simulations arereported. The first set of data consists of 18 simulations for T > T c in the rangeβ = 0.650 to 0.686 ≈ 0.99β c, with typically about 105 (almost uncorrelated)measurements. The correlation length varies in this β -range from ξ ≈ 3 to ξ ≈ 12(see Fig. 11 below). Here the use of improved estimators was very useful and ledto a further reduction of the statistical errors by a factor of about 2

−3.

The second set of simulations was performed near criticality. For each latticesize typically three independent runs with more than 105 measurements eachat different β values around β c were combined using the optimized reweightingtechnique, which is the most important additional tool in the finite-size scalingregion.




5.1 Simulations for T >∼

T c

The conceptually easiest way to measure critical exponents are simulations in

the high-temperature phase. In principle one simply has to fit the MC datafor ξ , C , m, χ, . . . with the expected power-law divergencies (10), (12)-(14) atcriticality. For high-precision estimates, however, the procedure is far from be-ing trivial. The problem is to locate the temperature range in which a simplepower-law Ansatz like (10) is valid. Clearly, since the omitted correction termsare positive powers of T /T c − 1, at first sight, one would like to perform thesimulations as close to T c as possible. But very close to T c the correlation lengthgets very large, and on finite lattices one starts seeing finite-size corrections.The only way around these correction terms are large enough lattice sizes. Inmany models one finds empirically that the thermodynamic limit is approachedwhen the linear lattice size satisfies L >∼ (6 − 8)ξ . But since the amplitude ξ o+ isnon-universal, this estimate is not guaranteed to be always true. Therefore thisquestion must be investigated very carefully for each model separately. With in-creasing temperature the correlation length decreases and finite-size correctionsare no longer a problem, but then it is for a different reason again not clear if the simple power-law Ansatz is valid. Very far away from T c the lattice structurebecomes important and the observables show a completely different behaviour.In an intermediate range one sees the confluent and analytic correction termswhich are very difficult to take into account in the fits. So in essence the problemis to locate a temperature window in which 1 ≪ ξ ≪ L.

There are many ways to extract the correlation length ξ from the asymptoticdecay of the spatial correlation function,

G(xi − xj) = σi · σj ∝ exp(−|xi − xj |/ξ ) . (50)

One way is measuring the Fourier transform, G(k) = xG(x) exp(

−ik

·x), for

a few long-wavelength modes and performing least-square fits to

G(k)−1 = c

3i=1

2(1 − cos ki) + (1/ξ )2

≈ c

k2 + (1/ξ )2

, (51)

where c is a constant and ki = (2π/L)ni, ni = 1, . . . , L. Recall that for zeromomentum the susceptibility is recovered, G(k = 0) = χ′/β . As an exampleFig. 10 shows the data for the 3D Heisenberg model [ n = (0,0,0), (1,0,0), (1,1,0),(1,1,1), (2,0,0), and (2,1,0)] at β = 0.686 ≈ 0.99β c. By repeating this analysisfor different temperatures one obtains the data shown in Fig. 11. The solid linesare fits according to

ξ (T ) = ξ 0+(T /T c − 1)−ν , (52)

with

β c = 0.69281 ± 0.00004 , (53)

ν = 0.698 ± 0.002 , (54)

ξ 0+ = 0.484 ± 0.002 , (55)



24 WolfhardJanke

0.000 0.005 0.010 0.015 0.020

Σi 2(1 - cos k

i)

0.001

0.003

0.005

0.007

G ( k ) - 1

V = 100

3

β = 0.686

ξ = 12.093

Fig. 10. Fit to the inverse of the Fourier transformed correlation function to computethe correlation length ξ .

and a goodness-of-fit parameter Q = 0.92, and

χ′(β )/β = χ′0(1 − β/β c)−γ , (56)

with

β c = 0.69294 ± 0.00003 , (57)

γ = 1.391

±0.003 , (58)

χ′0 = 0.955 ± 0.006 , (59)

and Q = 0.93. Notice that the correlation length in (52) is written as a functionof T and the susceptibility in (56) as a function of β . The high quality of thefits a posteriori justifies this choice and indicates that in the chosen temperaturerange confluent as well as analytic correction terms are negligible. If we rewriteT /T c − 1 = 1 − β/β c + (1 − β/β c)2 + . . . and consider ξ (β ) instead of ξ (T ), weexpect (and indeed confirmed) an analytical correction to asymptotic scaling.There is, however, so far no theoretical understanding why for the particularchoice of arguments the analytical correction terms should vanish.

5.2 Simulations near T c

The second set of data consists of simulations near T c on lattices of size up to483 (Holm and Janke 1993a, 1993b). In a later study focussing on topologicaldefects the maximal size could even be increased to 803 (Holm and Janke 1994).In the vicinity of T c, finite-size corrections are dominant and one has to employ




0.65 0.66 0.67 0.68 0.69

β

10

ξ

3D Heisenberg

Correlation Length

0.65 0.66 0.67 0.68 0.69

β

10

100χ’

3D Heisenberg

Susceptibility

Fig. 11. Scaling behaviour of the correlation length and susceptibility. The solid linesare fits to ξ (T ) and χ′(β ) according to the asymptotic power laws.

finite-size scaling (FSS) concepts to analyze the data. Usually one starts withan analysis of ratios of magnetization moments (Binder 1981), e.g.,

U L(β ) = 1 − 1

3

m4m22 , (60)

which leads to estimates of β c and the critical exponent ν . In the spontaneouslybroken low-temperature phase the magnetization distribution develops a double-peak structure with peaks at ±m0 = 0. Since the width of the peaks decreaseswith increasing lattice size one expects that U L approaches 2

3 for all T < T c.

In the disordered high-temperature phase the magnetization vanishes and themoments are determined by fluctuations alone. In the infinite volume limit thefluctuations become Gaussian and a simple calculation yields U L −→ 2(n −1)/3n = 4/9 for n = 3. Only at the transition point one expects a non-triviallimiting value which has been estimated by field theoretical methods (Brezinand Zinn-Justin 1985) to be U ∗ = 0.59684 . . . for n = 3. For finite systems,FSS predicts that the curves U L(β ) for different L intersect around (β c, U ∗)with slopes

∝L1/ν , apart from confluent corrections explaining small systematic

deviations. This allows an almost unbiased estimate of β c, U ∗ and the criticalexponent of the correlation length ν .

The data for the 3D Heisenberg model in Fig. 12 clearly confirm the theoret-ical expectations with a pronounced intersection point at (β c, U ∗) = (0.6930(1),0.6217(8)). The final numbers are actually obtained from a little more elaborate



26 WolfhardJanke

0 . 5 6

0 . 6 1

0 . 6 6

0 . 6 9 0 0 . 6 9 2 0 . 6 9 4 0 . 6 9 6

L=12

L=16

U

L=20

β

L=24

L=32

L=40

L=48

Fig. 12. The Binder parameter of the 3D Heisenberg model for various lattice sizes.The intersection points determine the inverse critical temperature β c = 0.6930(1).

analysis taking into account also the confluent corrections to asymptotic FSS(Holm and Janke 1993a, 1993b).

Also the slopes dU L/dβ at β = 0.6930 ≈ β c in Fig. 13 show the expectedbehaviour and a fit to the FSS prediction dU L/dβ ∝ L1/ν yields ν = 0.704(6),consistent with (54) and in very good agreement with estimates obtained by fieldtheoretical methods or from series expansions, cp. Table 2.

In the analysis of the magnetization and susceptibility one proceeds similarly.The fit to the magnetization data reweighted to β = 0.6930 ≈ β c shown inFig. 14 yields β/ν = 0.514(1), and from the FSS of the susceptibility one readsoff γ/ν = 1.9729(17). By multiplying the exponent ratios with the estimate of ν = 0.704(6), we finally arrive at the values for the critical exponents β and γ given in Table 2.

The analysis of the specific heat is much more complicated since the criticalexponent α is usually quite small and therefore the singularity in C not verypronounced. For the 3D Heisenberg model α is actually negative, so that we donot expect a divergence at all. By using the additional data on lattices up to 803

(Holm and Janke 1994), we obtained from the fit C = C reg + C 0Lα/ν shown inFig. 15 an estimate of α/ν = −0.225(80), resulting in α = −0.158(59). Due tothe rather large statistical error this estimate is still consistent with the value

obtained from hyperscaling, α = 2 − 3ν = −0.112(18). Actually a much moreprecise result was obtained from the corresponding FSS fit to the energy at β c.Using the Ansatz e = ereg + e0L(α−1)/ν , we obtained α/ν = −0.166(31), trans-lating into α = −0.117(23). Obviously this value is in a much better agreementwith the hyperscaling prediction.




0 . 0

0 . 5

1 . 0

1 . 5

1 . 0 1 . 2

l o g

d U / d β

1 . 4 1 . 6 1 . 8

1/slope = 0.704 ± 0.006 at β = 0.6930

log L

Fig. 13. FSS of the Binder parameter slopes at β = 0.6930 ≈ β c. The linear fit yieldsan estimate of the correlation length exponent, ν = 0.704(6).

- 0 . 9

- 0 . 8

- 0 . 7

- 0 . 6

- 0 . 5

- 0 . 4

1 . 0 1 . 2

l o g

M

1 . 4 1 . 6 1 . 8

slope=0.514 ± 0.001 at β = 0.6930

log L

Fig. 14. FSS of the magnetization at β = 0.6930 ≈ β c. The linear fit yields an estimateof the exponent ratio, β /ν = 0.514(1).



28 WolfhardJanke

0 20 40 60 80 100L

2.0

2.5

3.0

3.5

C

at β=0.6930

C(L) = 4.76 - 4.12 L-0.225

Fig. 15. FSS of the specific heat at β = 0.6930 ≈ β c. The non-linear fit takinginto account a regular background term yields an estimate of the exponent ratio,α/ν = −0.225(80).

Table 2. Critical coupling and critical exponents of the 3D classical Heisenberg (n = 3)model

method β c ν γ β α δ

g-expansiona - 0.705(3) 1.386(4) 0.3645(25) −0.115(9) 4.802(37)

ǫ-expansionb - 0.710(7) 1.390(10) 0.368(4) −0.130(21) 4.777(70)

MCc 0.6929(1) 0.706(9) 1.390(23) 0.364(7) −0.118(27) 4.819(36)

MCd 0.6930(1) 0.704(6) 1.389(14) 0.362(4) −0.112(18) 4.837(11)

MCe 0.693035(37) 0.7036(23) 1.3896(70) 0.3616(31) −0.1108(69) −

seriesf 0.6929(1) 0.712(10) 1.400(10) 0.363(10) −0.136(30) 4.86(10)

a Le Guillou and Zinn-Justin 1977, 1980b Le Guillou and Zinn-Justin 1985c Peczak et al. 1991d

Holm and Janke 1993a, 1993be Chen et al. 1993f Adler et al. 1993




6 Concluding Remarks

The intention of these lecture notes was to give an elementary introduction to the

basic concepts of modern Monte Carlo simulations and to illustrate their useful-ness by applications to one typical model. Since the choice of the 3D Heisenbergmodel was obviously biased by my own work in this field, I want to concludewith at least a few remarks on the 3D Ising and 3D XY model, for which alsoquite a few high-precision simulations have been performed.

Due to its relative simplicity, the 3D Ising model is the best studied modelof all O(n) spin systems. Apart from finite-size scaling analysis as describedhere, also many other techniques have been applied, including the Monte Carlorenormalization group (MCRG) and the finite-size scaling of partition functionzeros. This has led to quite a few very accurate estimates of critical exponents.Some of them are compiled in Table 3, where for comparison also field theoryand series expansion estimates are given. As an amusing side remark it is worth

mentioning the Rosengren (1986) conjecture that the critical coupling of the 3DIsing model is given by β c = tanh−1[(

√ 5 − 2) cos(π/8)] = 0.221 658 637 . . . –

a value which is indeed in impressive agreement with the most precise MonteCarlo estimates!

Table 3. Critical coupling and selected critical exponents of the 3D Ising ( n = 1)model

method β c ν γ

g-expansiona - 0.6300(15) 1.241(2)

ǫ-expansionb - 0.6310(15) 1.2390(25)

MCRGc 0.221 652(3) 0.624(2) -

MCd 0.2216595(26) 0.6289(8) 1.239(7)

seriese 0.221 655(5) 0.631(4) 1.239(3)

a Le Guillou and Zinn-Justin 1977, 1980b Le Guillou and Zinn-Justin 1985c Baillie et al. 1992d Ferrenberg and Landau 1991e Adler 1983

A similar accuracy could also be reached for the 3D XY model, the sim-

plest model of the O(2) universality class which governs the critical behaviourof the λ-transition in liquid helium. Some recent results of Monte Carlo simu-lations employing the Swendsen-Wang and Wolff cluster update algorithm, aswell as estimates using series expansions and field theory methods are compiledin Table 4.



30 WolfhardJanke

Table 4. Critical coupling and selected critical exponents of the 3D classical XY(n = 2) model

method β c ν γ

4He experimenta - 0.6705(6) -

g-expansionb - 0.669(2) 1.3160(25)

ǫ-expansionc - 0.671(5) 1.315(7)

MCd 0.45421(8) - 1.327(8)

MCe 0.45408(8) 0.670(2) 1.316(5)

MCf 0.45417(1) 0.662(7) 1.324(1)

seriesg 0.45406(5) 0.67(1) 1.315(9)

seriesh 0.45414(7) ≈0.673 ≈1.325

a Goldner and Ahlers 1992b Le Guillou and Zinn-Justin 1977, 1980c Le Guillou and Zinn-Justin 1985d Hasenbusch and Meyer 1990e Janke 1990f Gottlob and Hasenbusch 1993g Butera et al. 1993h Adler et al. 1993

By comparing the various Monte Carlo estimates collected in Tables 2-4 withresults from field theory and series expansions it is fair to conclude that for O( n)spin models modern Monte Carlo techniques are at the present time superior

to series expansion analyses. The recently derived critical exponents are in factcompetitive in accuracy with estimates obtained with the best and very elaboratemethods of field theory.

Acknowledgements

The results for the 3D Heisenberg model are taken from joint work with C. Holm.It is a pleasure to thank him for a very fruitful and enjoyable collaboration. I amalso grateful to S. Kappler and T. Sauer for numerous stimulating discussionsand collaborations on various aspects of Monte Carlo simulations described here.Along the way, discussions with B. Berg, D. Johnston, and R. Villanova and theircomments helped a lot. Last but not least I would like to thank K. Binder and

D.P. Landau for sharing their experience and insight in many of the problemstouched in these lecture notes. Finally, I gratefully acknowledge a Heisenbergfellowship by the Deutsche Forschungsgemeinschaft.




Appendix: Program Codes

The accompanying diskette contains five FORTRAN codes (is clu.f, reweight.f,

rew his.f, 1dis ex.f, 2disnm ex.f) and two data files (3d e004.plo, 3d c004.plo)which can be used to reproduce Figs. 3, 7, and 8, and to generate an Ising modelanalog of Fig. 9. The output files denoted by ...plo are kept as simple as possibleto allow easy plotting, e.g. with the standard utility gnuplot .

is clu.f is a Monte Carlo simulation program for the nearest-neighbor Isingmodel with subroutines for the Wolff single-cluster (sc), Swendsen-Wang multiple-cluster (sw), Metropolis (me), and heat-bath (hb) update algorithm, which canbe selected in the main parameter statement. All subroutines are set up towork for general D-dimensional (hyper-) cubic lattices of size LD with periodicboundary conditions. The dimension, lattice size, simulation temperature, andthe statistics parameters are also defined in the main parameter statement (thedimension and lattice size have to be changed globally in all subroutines as well).With the choice of parameters as given in is clu.param example , the 2D Ising(multiple-cluster) Monte Carlo data of Figs. 7 and 8 can be reproduced withinabout one minute run-time. Of course the detailed timing depends on the type of PC or workstation, but it should always be possible to run the simulation inter-actively. The average energy, magnetization, specific heat, susceptibility, Binderparameter, and cluster averages are written on standard output. Furthermore,the energy and magnetization histograms, and the “microcanonical” magnetiza-tion averages |m| and m2 (cp. (46)) at the simulation temperature are savedin the files ehis b0.plo, mhis b0.plo , malis b0.plo, and m2lis b0.plo for easy plot-ting, and in e b0.his for further reweighting analyses (containing the necessaryparameter informations). With these output files it is straightforward to produce

for the Ising model a plot similar to Fig. 9. If desired the time evolution of theenergy and magnetization measurements can be saved in the files e series.plo

and m series.plo, respectively, by turning on the corresponding logical switchesin the parameter statement.

For simplicity the standard UNIX random number generator RAND( ) iscalled in the MC program. For illustration purposes this generator is goodenough, but for a serious simulation study it should at any rate be replacedby a more reliable random number generator. Again only for simplicity all sta-tistical error analysis subroutines are omitted. For a sensible MC study this isclearly unacceptable.

reweight.f takes as input the energy histogram and the “microcanonical” mag-

netization lists stored in e b0.his and computes by reweighting the energy andspecific heat, the susceptibilities χ/β and χ ′/β , and the Binder parameter as afunction of inverse temperature β . The desired β -range can be set in the param-eter statement, and the dimension and lattice size parameters must be the sameas in is clu.f . The results are written into the files e016 mc.plo , c016 mc.plo,



32 WolfhardJanke

sus016 mc.plo, chi016 mc.plo, and U016 mc.plo, respectively, where 016 indi-cates the linear lattice size.

The MC data for the one-dimensional Ising chain can be tested against the

exact results provided by 1dis ex.f for any chain length. The two-dimensionalMC results can be used together with the output from 2disnm ex.f to reproduceFig. 7. Further comparison data for a 162 lattice from high-statistics single-cluster simulations at β c = ln(1+

√ 2)/2 are 2+e = 0.546 85(10), C = 1.4978(10),

χ = 139.669(31), |C | = 139.656(29), and U = 0.611 537(50). In three dimen-sions the MC data can be compared with the exact energy and specific heatcurves for a 43 lattice contained in the data files 3d e004.plo and 3d c004.plo.

rew his.f reads again as input the energy histogram stored in e b0.his , computesreweighted histograms, and stores them in, e.g., ehis b4750.plo. The dimensionand lattice size parameters must be the same as in is clu.f . Here b4750 indicatesthat the histogram is reweighted to β = 0.4750. The new inverse temperature is

inquired interactively by the program. In this way Fig. 8 can be reproduced. If gnuplot is used for plotting the histograms, then by using the escape character”!” the reweighting program can be called and the new histogram immediatelydisplayed without leaving the plot session.

1dis ex.f computes the exact temperature dependence of the energy, specificheat, and susceptibility of the one-dimensional Ising chain with periodic bound-ary conditions of arbitrary length L. For e.g. L = 16, the results are stored inthe files 1d e016.plo, 1d c016.plo , and 1d chi016.plo.

2disnm ex.f implements the exact solution of Ferdinand and Fisher (1969)

for the 2D nearest-neighbor interaction Ising model on finite Lx × Ly lattices(Lx, Ly = even) with periodic boundary conditions. The desired lattice size andinverse temperature range can be chosen in the parameter statement of themain program. The output are two data files, e.g. 2d e016.plo and 2d c016.plo

for Lx = Ly = 16, containing minus the internal energy per site, −E/V , and thespecific heat per site, C , as a function of the inverse temperature β . By runningthis code for various lattice sizes, Fig. 3 can be reproduced.




References

Adler, J. (1983): J. Phys. A16, 3585

Adler, J., Holm, C., Janke, W. (1993): Physica A201, 581Adler, S.L. (1988): Phys. Rev. D37, 458Alves, N.A., Berg,B.A., Sanielevici, S. (1992): Nucl. Phys. B376, 218Baillie, C.F. (1990): Int. J. Mod. Phys. C1, 91Baillie, C.F., Gupta, R., Hawick, K.A., Pawley, G.S. (1992): Phys. Rev. B45, 10438Barber, M.E. (1983): in Phase Transitions and Critical Phenomena , Vol. 8, edited by

C. Domb and J.L. Lebowitz (Academic, New York)Berg,B.A., Neuhaus,T. (1991): Phys. Lett. B267, 249Berg,B.A., Neuhaus,T. (1992): Phys. Rev. Lett. 68, 9Berezinskii, V.L. (1971): Zh. Eksp. Teor. Fiz. 61, 1144 [Sov. Phys.–JETP 34, 610

(1972)]Berlin, T.H., Kac, M. (1952): Phys. Rev. 86, 821Billoire,A., Lacaze, R., Morel,A., Gupta,S., Irback, A., Petersson, B. (1991): Nucl.

Phys. B358, 231Binder, K., (1979): in Monte Carlo Methods in Statistical Physics , edited by K. Binder(Springer, Berlin)

Binder, K., (1981): Z. Phys. B43, 119Binder, K., (1982): Phys. Rev. A25, 1699Binder, K., (1987): Rep. Prog. Phys. 50, 783Binder, K. (ed.), (1992a): The Monte Carlo Method in Condensed Matter Physics

(Springer, Berlin)Binder, K. (1992b): in Computational Methods in Field Theory , p. 59, Schladming 1992

lecture notes, edited by H. Gausterer and C.B. Lang (Springer, Berlin)Binder, K., Heermann, D.W. (1988): Monte Carlo Simulations in Statistical Physics:

An Introduction (Springer, Berlin)Binder, K., Landau, D.P. (1984): Phys. Rev. B30, 1477Borgs, C., Janke,W. (1992): Phys. Rev. Lett. 68, 1738

Borgs, C., Kotecky, R. (1990): J. Stat. Phys. 61, 79Borgs, C., Kotecky, R. (1992): Phys. Rev. Lett. 68, 1734.Borgs, C., Kotecky, R., Miracle-Sole, S. (1991): J. Stat. Phys. 62, 529.Brezin, E., Zinn-Justin,J. (1985): Nucl. Phys. B257[FS14], 867Butera, P., Comi, M., Guttmann, A.J. (1993): Phys. Rev. B48, 13987Challa, M.S.S., Landau, D.P., Binder, K. (1986): Phys. Rev. B34, 1841Chen, K., Ferrenberg, A.M., Landau, D.P. (1993): Phys. Rev. B48, 3249Creutz, M. (1987): Phys. Rev. D36, 515Dimitrovic, I., Hasenfratz, P., Nager, J., Niedermayer, F. (1991): Nucl. Phys. B350, 893Doll, J.D, Coalsen, R.D., Freeman, D.L. (1985): Phys. Rev. Lett. 55, 1Efron, B. (1982): The Jackknife, the Bootstrap and other Resampling Plans (SIAM,

Philadelphia)Ferdinand, A.E., Fisher,M.E. (1969): Phys. Rev. 185, 832

Ferrenberg, A.M., Landau, D.P. (1991): Phys. Rev. B44

, 5081Ferrenberg, A.M., Swendsen,A.M. (1988): Phys. Rev. Lett. 61, 2635Ferrenberg, A.M., Swendsen,A.M. (1989a): Phys. Rev. Lett. 63, 1658(E)Ferrenberg, A.M., Swendsen,A.M. (1989b): Phys. Rev. Lett. 63, 1195Ferrenberg, A.M., Landau, D.P., Binder, K. (1991): J. Stat. Phys. 63, 867Fortuin, C.M., Kasteleyn, P.W. (1972): Physica 57, 536



34 WolfhardJanke

Fisher, M.E., Berker, A.N. (1982): Phys. Rev. B26, 2507Fortuin, C.M. (1972a): Physica 58, 393Fortuin, C.M. (1972b): Physica 59, 545

Goldner, L.S., Ahlers, G. (1992): Phys. Rev. B45, 13129Goodman, J., Sokal, A.D. (1986): Phys. Rev. Lett. 56, 1015Goodman, J., Sokal, A.D. (1989): Phys. Rev. D40, 2035Gottlob, A.P., Hasenbusch,M. (1993): Physica A201, 593Gunton, J.D., Miguel, M.S., Sahni, P.S. (1983): in Phase Transitions and Critical Phe-

nomena , Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York)Gupta, R., DeLapp, J., Battrouni, G.G., Fox, G.C., Baillie, C.F., Apostolakis, J. (1988):

Phys. Rev. Lett. 61, 1996Hammersley, J.M., Handscomb, D.C. (1965): Monte Carlo Methods (London)Hasenbusch, M. (1990): Nucl. Phys. B333, 581Hasenbusch, M., Meyer, S. (1990): Phys. Lett. B241, 238Heermann, D.W. (1990): Computer Simulation Methods in Theoretical Physics , 2nd

ed., (Springer, Berlin)

Heermann, D.W., Burkitt, A.N. (1990): Physica A162, 210Herrmann,H.J., Janke, W., Karsch, F. (eds.) (1992): Dynamics of First Order Phase

Transitions (World Scientific, Singapore)Holm, C., Janke,W. (1993a): Phys. Lett. A173, 8Holm, C., Janke, W. (1993b): Phys. Rev. B48, 936Holm, C., Janke, W. (1994): J. Phys. A27, 2553Ito, N., Kohring, G.A. (1993): Physica A201, 547Janke, W. (1990): Phys. Lett. A148, 306Janke, W. (1993): Phys. Rev. B47, 14757Janke, W. (1994): Recent Developments in Monte Carlo Simulations of First-Order

Phase Transitions , in Computer Simulations in Condensed Matter Physics VII ,p. 29, eds. D.P. Landau, K.K. Mon, and H.-B. Schuttler (Springer, Berlin)

Janke, W., Kappler, S. (1994): Nucl. Phys. B (Proc. Suppl.) 34, 674

Janke, W., Kappler, S. (1995a): Phys. Lett. A197, 227Janke, W., Kappler, S. (1995b): Nucl. Phys. B (Proc. Suppl.) 42, 770Janke, W., Kappler, S. (1995c): Europhys. Lett. 31, 345Janke, W., Kappler, S. (1995d): Mainz preprint KOMA-95-65 (September 1995), hep-

lat/9509057, to appear in the proceedings of the conference Lattice ’95 , Melbourne,Australia, July 11 – 15, 1995

Janke, W., Kappler, S. (1995e): Nucl. Phys. B (Proc. Suppl.) 42, 876Janke, W., Kappler, S. (1995f): Phys. Rev. Lett. 74, 212Janke, W., Kleinert, H. (1986): Phys. Rev. B33, 6346Janke, W., Sauer, T. (1994): Phys. Rev. E49, 3475Janke, W., Sauer, T. (1995): J. Stat. Phys. 78, 759Janke, W., Berg,B.A., Katoot, M. (1992): Nucl. Phys. B382, 649Kandel, D., Domany, E., Brandt,A. (1989): Phys. Rev. B40, 330Kandel, D., Domany, E., Ron, D., Brandt, A., Loh, Jr., E. (1988): Phys. Rev. Lett. 60,

1591Kappler, S. (1995): Ph.D. Thesis, Johannes Gutenberg-Universitat MainzKasteleyn, P.W., Fortuin, C.M. (1969): J. Phys. Soc. Japan 26 (Suppl.), 11Kleinert, H. (1989a): Gauge Fields in Condensed Matter , Vol. I (World Scientific, Sin-

gapore)




Kleinert, H. (1989b): Gauge Fields in Condensed Matter , Vol. II (World Scientific, Sin-gapore)

Kosterlitz,J.M. (1974): J. Phys. C7, 1046

Kosterlitz, J.M., Thouless, D.J. (1973): J. Phys. C6, 1181Lee, J., Kosterlitz, J.M. (1990): Phys. Rev. Lett. 65, 137Lee, J., Kosterlitz, J.M. (1991): Phys. Rev. B43, 3265Le Guillou, J.C., Zinn-Justin, J. (1977): Phys. Rev. Lett. 39, 95Le Guillou, J.C., Zinn-Justin, J. (1980): Phys. Rev. B21, 3976Le Guillou, J.C., Zinn-Justin, J. (1985): J. Physique Lett. 46, L137Mack, G. (1988): in Nonperturbative quantum field theory , Cargese lectures 1987, edited

by G.‘t Hooft et al. (Plenum, New York)Mack, G., Meyer, S. (1990): Nucl. Phys. B (Proc. Suppl.) 17, 293Madras, N., Sokal, A.D. (1988): J. Stat. Phys. 50, 109Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.W., Teller, E. (1953): J.

Chem. Phys. 21, 1087Miller, R.G. (1974): Biometrika 61, 1Neuberger, H. (1988): Phys. Lett. B207, 461Patrascioiu, A., Seiler, E. (1995): Munich preprint MPI-PhT/95-73, hep-lat/9508014Peczak, P., Landau, D.P. (1990a): Bull. Am. Phys. Soc. 35, 255Peczak, P., Landau, D.P. (1990b): J. Appl. Phys. 67, 5427Peczak, P., Landau, D.P. (1993): Phys. Rev. B47, 14260Peczak, P., Ferrenberg, A.M., Landau, D.P. (1991): Phys. Rev. B43, 6087Potts, R.B. (1952): Proc. Camb. Phil. Soc. 48, 106Privman, V. (editor) (1990): Finite-Size Scaling and Numerical Simulations of Statis-

tical Systems (World Scientific, Singapore)Privman, V., Fisher,M.E. (1983): J. Stat. Phys. 33, 385Privman, V., Rudnik, J. (1990): J. Stat. Phys. 60, 551Rosengren,A. (1986): J. Phys. A19, 1709Rummukainen, K. (1993): Nucl. Phys. B390, 621Sauer, T. (1994): Ph.D. Thesis, Freie Universitat Berlin

Sokal, A.D. (1989): Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms , Cours de Troisieme Cycle de la Physique en Suisse Romande, Lau-sanne

Sokal, A.D. (1992): Bosonic Algorithms , in Quantum Fields on the Computer , p. 211,edited by M. Creutz (World Scientific, Singapore)

Swendsen, R.H., Wang, J.-S. (1987): Phys. Rev. Lett. 58, 86Swendsen, R.H., Wang, J.-S., Ferrenberg, A.M. (1992): in The Monte Carlo Method in

Condensed Matter Physics , edited by K. Binder (Springer, Berlin)Tamayo,P. (1993): Physica A201, 543Wolff, U. (1989a): Phys. Rev. Lett. 62, 361Wolff, U. (1989b): Nucl. Phys. B322, 759Wolff, U. (1989c): Phys. Lett. A228, 379Wolff, U. (1990): Nucl. Phys. B334, 581Wu, F.Y. (1982): Rev. Mod. Phys. 54, 235Wu, F.Y. (1983): Rev. Mod. Phys. 55, 315(E)



2D Ising model exact solution, 7, 19

Binder parameter, 25

critical exponent, 5

critical slowing down, 11

energy distribution, 18exponential autocorrelation time, 9exponential slowing down, 17

finite-size scaling Ansatze, 7first-order phase transition, 4, 17

importance sampling, 8improved cluster estimator, 15integrated autocorrelation time, 9

Markov chain, 8

microcanonical averages, 20multiple-cluster update, 13

non-local update algorithms, 11

O(n) spin models, 2

second-order phase transition, 5single-cluster algorithm, 13statistical error, 10

universality, 5

36

monte carlo simulation of spin system

Documents